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Experimental investigation on the aerodynamic behavior of square cylinders with rounded corners
Journal of Fluids and Structures 44 (2014) 195–204
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Journal of Fluids and Structures journal homepage: www.elsevier.com/locate/jfs
Experimental investigation on the aerodynamic behavior of square cylinders with rounded corners Luigi Carassale n, Andrea Freda, Michela Marrè-Brunenghi DICCA – Department of Civil, Chemical and Environmental Engineering, University of Genova, Italy
a r t i c l e i n f o
abstract
Article history: Received 8 January 2013 Accepted 10 October 2013 Available online 16 November 2013
The influence of corner shaping on the aerodynamic behavior of square cylinders is studied through wind tunnel tests. Beside the sharp-edge corner condition considered as a benchmark, two different rounded-corner radii (r/b¼ 1/15 and 2/15) are studied. Global forces and surface pressure are simultaneously measured in the Reynolds number range between 1.7 104 and 2.3 105. The measurements are extended to angles of incidence between 01 and 451, but the analysis and the discussion presented herein is focused on the low angle of incidence range. It is found that the critical angle of incidence, corresponding to the flow reattachment on the lateral face exposed to the flow, decreases as r/b increases and that an intermittent flow condition exists. In the case of turbulent incoming flow, two different aerodynamic regimes governed by the Reynolds number have been recognized. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Square cylinders Rounded corners Intermittence Critical angle Reynolds number effect
1. Introduction A careful modeling of the corners shape of building and structural elements has become a major objective for a windresponse-oriented optimal design. The introduction of rounded or chamfered corners has often the positive effect of reducing the drag force and the fluctuation of the transversal force due to vortex shedding (e.g. Kwok et al., 1988; Tamura and Miyagi, 1999); on the other hand, it can produce a more complicated aerodynamic behavior whose physical or numerical modeling may be challenging at design stage for wind engineers. As a matter of fact, in contrast to sharp-edge bodies, the rounding or the chamfering of the corners may lead to the absence of fixed separation points, and a significant dependency on both the Reynolds number and the characteristics of the incoming flow (e.g. Delany and Sorensen, 1953; Tamura and Miyagi, 1999; Tamura et al., 1998; Larose and D'Auteuil, 2008). The square cylinder is probably the simplest sharp-edge body and probably the most commonly investigated in aerodynamics; it is therefore the natural candidate to investigate the effect of corner shaping on the aerodynamic behavior of bluff bodies. With the twofold purpose of investigating the basic behavior of rounded corners and providing technical information useful for wind engineers, a wind-tunnel tests campaign has been carried out on square cylinders with rounded corners. Beside the sharp-corner case, rigid models of cylinders with two corner radii (r/b¼1/15 and 2/15) have been realized and tested. The global forces and the pressure field along the mid-span cross-section have been measured in the Reynolds number range between 1.7 104 and 2.3 105. Two levels of turbulence intensity (0.2% and 5%) have been considered. Section 2 provides a brief review of the current knowledge on the aerodynamic behavior of sharp-edge square cylinders, with particular reference to the qualitative modification of the flow field with the variation of angles of incidence.
n
Corresponding author. Tel.: þ39 010 353 2226. E-mail address: [email protected] (L. Carassale).
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These concepts are then used as a guide for the interpretation of the experimental results presented in Section 3 and discussed in Section 4. The discussion is focused on two specific phenomena: (1) the intermittence of the vortex-shedding regime at the critical angle of incidence in smooth flow and (2) the transition from subcritical to supercritical Reynolds number regime in turbulent flow, with the inversion of the steady lift coefficient slope at zero angle of incidence.
2. Aerodynamic behavior of sharp-corner square cylinders: the dependency on the angle of incidence Sharp-edge square cylinders have been studied by several researchers who focused their attention on numerous aerodynamic parameters including pressure distribution, drag and lift forces, vortex shedding properties, as well as near wake velocity field (e.g. Vickery, 1966; Lee, 1975; Okajima, 1982; Igarashi, 1984; Nakamura and Ohya, 1984). The flow pattern around a square cylinder is strongly dependent on the angle of incidence α, and at least two characteristic flow regimes are clearly identified and separated by a critical angle αcr located about 12–151 (Igarashi, 1984). The boundary layer is completely separated from both the lateral faces for α o αcr, whereas the flow reattaches on the side exposed to the wind forming a separation bubble for α 4 αcr (e.g. Huang et al., 2010; Huang and Lin, 2011). A further sub-classification of the two regimes could be mentioned (Igarashi, 1984), but is not relevant in the present context. The subcritical regime (α o αcr) is characterized by negative slope of the lift coefficient, which sharply changes to positive as α becomes greater than αcr. Besides, the transition from the subcritical to the supercritical regime produces a rapid increment of the Strouhal number that corresponds to the reduction of the wake width due to the flow reattachment (Lee, 1975). In the critical regime both the drag coefficient and the fluctuating lift coefficient have a minimum value. This scenario is somehow dependent on the Reynolds number (Yen and Yang, 2011), as well as on the characteristics of the incoming flow. In particular, the thickening of the shear layers due to a small-scale free-stream turbulence promotes the formation of the separation bubble, which tends to appear for smaller angle of incidence and to shrink towards the leading edge (Lee, 1975).
3. Experimental results The experimental tests have been carried out in the closed-circuit wind tunnel at the University of Genova, whose cross section is 1640 1350 mm2. The models have span length l ¼500 mm; they are fabricated through the assemblage of aluminum plates and machined to reduce geometrical imperfections below 0.1% of the cross-section size b. The angle of incidence of the prisms is measured through a digital protractor with resolution 0.11. End plates are installed at the extremities of the models. A force balance realized by six resistive load cells is employed to measure global forces. The midspan cross section of the models is instrumented by a ring of N pressure taps (N ranges from 20 to 44 for different models) connected through short tubes to pressure scanners mounted inside the model. Beside the sharp-edge square cross section used as a benchmark test, two rounded-corner configurations with r/b¼1/15 and 2/15 (r being the corner radius) are considered. Fig. 1 shows the cross section of the tested cylinders and the reference system used for the presentation of the results. The force balance measurements are analyzed calculating the steady aerodynamic drag and lift coefficients (CD and CL), the Strouhal number (St) and the fluctuating lift coefficient (C~ L ) defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E½L2 E½L2 E½D E½L ns b ; C~ L ¼ CD ¼ ; CL ¼ ; St ¼ ; ð1Þ 2 2 U 0:5ρblU 0:5ρblU 0:5ρblU 2 where D and L are, respectively, the measured drag and lift forces (Fig. 1); ρ is the air density estimated on the basis of the temperature measured inside the wind tunnel; U is the free stream mean velocity measured through a static-Pitot tube installed upstream the test section; E[] is the statistic average operator that is implemented as a time average adopting the hypothesis of ergodic behavior; ns is the vortex shedding peak frequency. The vortex shedding frequency is estimated by fitting the power spectral density function of L through a Gaussian function in the neighborhood of the vortex shedding peak. The spectral analysis of L was carried out separately for the two concurrent flow configurations when a clear intermittent behavior was observed. Similar analyses carried out on the velocity recorded in the wake were used for verification purpose.
Fig. 1. Experimental setup.
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Analogously, the steady pressure coefficient is defined as Cp ¼
E½p p0 0:5ρU 2
;
ð2Þ
where p is the pressure measured on the body surface and p0 is the static wind-tunnel pressure. The influence of the Reynolds number is investigated varying both the wind-tunnel velocity U in the range 5–25 m/s, and the body size b in the range 50–150 mm. The Reynolds number is defined as Re ¼Ub/ν, ν being the kinematic viscosity, and considering the cross-section size as reference also for rotated configurations. The kinematic viscosity was calculated on the basis of the air temperature measured in the test section. No blockage correction is adopted since the primary purpose of the paper is the comparative analysis of the aerodynamic behavior of bodies with similar shape and the qualitative description of some observed fluid-dynamic phenomena. Besides, since these phenomena are related to strong modifications of the flow topology, the use of blockage-correction rules should be necessarily restricted to flow configurations that are far enough from transition points. This would introduce artificial discontinuities in the experimental data that would complicate interpretation. On the other hand, it is important to mention that several models with different size (corresponding to the ratios b/B from 2.5% to 11%, B being the size of the wind tunnel cross-section in direction orthogonal to the prism axis) have been tested and it emerged that the quality of the observed phenomena remains very consistent. Two conditions of incoming flow are considered: (1) smooth flow characterized by a longitudinal turbulence intensity Iu about 0.2% and (2) turbulent flow, produced through a grid realized by square bars, characterized by Iu about 5% and integral length scale Lu about 20 mm. Fig. 2 shows the variation of CD (a), CL (b), St (c) and C~ L (d) with respect to the angle of incidence for the three considered models in smooth flow. No significant influence of Re is found in the whole range explored during the experiments. It can be observed that these three models have a similar qualitative behavior characterized by the inversion of the lift slope at the critical angle of incidence αcr. For α ¼ αcr the steady drag and the fluctuating lift coefficients have a minimum value, while St increases sharply. The smooth fitting of the experimental data is interrupted at α ¼ αcr where a discontinuity exists 2.4
0.4
2.2
0.2 0 -0.2
CL
CD
2 1.8
-0.4 1.6
-0.6
1.4 1.2
-0.8 0
10
20
30
40
50
-1
0
10
α [°]
20
30
40
50
30
40
50
α [°]
0.18
1.2
0.17
1
0.16
0.8
0.15
St
0.6 0.14 0.4
0.13
0.2
0.12 0.11
0
10
20
α [°]
30
40
50
0
0
10
20
α [°]
Fig. 2. Steady drag (a) and lift (b) coefficients, Strouhal number (c) and fluctuating lift coefficient (d) in smooth flow as functions of α. Re¼3.7 104 for r ¼0; Re¼ 2.7 104 for r/b¼ 1/15 and r/b¼ 2/15.
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2.4
0.8 0.6
2.2
0.4 0.2
1.8
CL
CD
2
0 -0.2
1.6
-0.4
1.4
-0.6 1.2 1
-0.8 0
10
20
30
40
50
-1
0
10
α [°]
20
30
40
50
30
40
50
α [°] 1.2
0.2 0.19
1
0.18 0.8
St
0.17 0.16
0.6
0.15 0.4
0.14 0.13
0.2
0.12 0.11
0
10
20
α [°]
30
40
50
0
0
10
20
α [°]
Fig. 3. Steady drag (a) and lift (b) coefficients, Strouhal number (c) and fluctuating lift coefficient (d) in turbulent flow as functions of α. Re¼3.6 104 for r¼ 0; Re¼7.9 104 for r/b¼ 1/15; Re¼2.5 104 for r/b ¼ 2/15.
(see Section 4 for a discussion about this issue). The value of αcr decreases as r/b increases; for the sharp-corner model αcr has been found to be about 121, which is in substantial accord to previous experimentations (see Huang and Lin, 2011, for a review of previous results). In the case of rounded-corner models αcr is about 71 and 51 for r/b ¼1/15 and 2/15, respectively. Tamura and Miyagi (1999), working in similar flow conditions, found αcr ¼41 for r/b¼2.5/15. Fig. 3 shows the variation of CD (a), CL (b), St (c) and C~ L with respect to the angle of incidence, in turbulent flow at Re¼3.6 104 for r ¼0, Re¼7.9 104 for r/b¼ 1/15 and Re¼ 2.5 104 for r/b¼ 2/15. The qualitative comparison with the results presented in Fig. 2 shows that the increment of the free stream turbulence produces a slight reduction of the critical angle (for all the models) and promotes a smooth transition from subcritical to supercritical angle regimes. As far as the sharp-corner model and the rounded-corner model with r/b¼1/15 are concerned, the quantities shown in Fig. 3 are practically invariant through the whole observed range of Re (Re ¼1.7 104–1.6 105 for sharp-corner model and Re¼2.5 104–2.3 105 for r/b¼1/15). On the contrary, the model with r/b¼2/15 showed a strong dependency on Re, in particular for small angles of incidence. Fig. 4 shows CD (a), dCL/dα (b), St (c) and C~ L (d) measured in turbulent flow for α ¼ 0 on the rounded-corner model with r/b¼ 2/15. It can be observed that a drag crisis appears for Re between 5 104 and 1.2 105. Across this critical regime the drag coefficient drops from about 1.25 to 0.80; the lift slope passes from about 10 to þ 5 reaching values about 15 during the transition; St practically doubles jumping from 0.13 to 0.25; the fluctuating lift decreases significantly during the transition. Between Re¼6.6 104 and 8.6 104 no regular vortex shedding was observed. Delany and Sorensen (1953) carried out experiments on rounded-corner square cylinders in smooth flow finding a drag crisis at Re⋍6 105 for r/b¼2.5/15 and Re⋍2.5 105 for r/b¼5/15 (in the mentioned reference Re is defined consistently with the definition adopted herein). Fig. 5 shows the variation of CD (a), CL (b), St (c) and C~ L (d) with respect to the angle of incidence, for r/b¼2/15, in turbulent flow at different Re between 4.90 104 and 1.81 105. The transition observed in Fig. 4 reflects into the existence of two distinct behaviors referred to as subcritical (Re oRecr⋍7.5 104) and supercritical regime (Re4Recr). In the subcritical regime the drag coefficient for α ¼ 0 is about 1.3, (slightly lower than in smooth flow). The minimum value of the steady lift coefficient is about 0.7, and is obtained for α ¼3–41 (slightly before than αcr in smooth flow). The Strouhal number is about 0.13 for α ¼ 0 and rapidly increases up to 0.18 for α ¼41 (again not far from the values observed in smooth flow). As Re increases, the steady lift slope for α ¼0 switches from negative to positive. Further increments of Re have the result of extending the range of α where the slope is positive. The maximum value of the steady lift coefficient increases as
L. Carassale et al. / Journal of Fluids and Structures 44 (2014) 195–204
199
10
1.4 1.3
5
dCL/dα
CD
1.2 1.1 1
0
-5
0.9 -10 0.8 0.7
0
0.5
1
1.5
2
x 105 2.5
-15
0
0.5
1
Re
1.5
2
x 105 2.5
1.5
2
x 105 2.5
Re 0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
St
0.35
0.1
0
0.5
1
1.5
2
x 105 2.5
Re
0.1
0
0.5
1
Re
Fig. 4. Steady drag (a) and lift (b) coefficients, Strouhal number (c) and fluctuating lift coefficient (d) for r/b¼ 2/15 in turbulent flow at α ¼ 01 as functions of Re.
Re increases and is obtained for larger and larger angles of incidence. The maximum observed CL is about 0.45 and appears for α ¼41 at Re¼1.81 105. In the region where the steady lift slope is positive St is about 0.27 (approximately twice the value obtained in smooth flow) and is practically insensitive to Re. As the slope of CL becomes negative, the regular vortex shedding disappears, until both CL and St recover the trend observed in subcritical regime; in the observed range of Re it happens before α ¼101. The minimum value of the drag coefficient is reached for the angle of incidence corresponding to the maximum values for CL. The results shown in Figs. 4 and 5 are obtained using a model with b¼150 mm. The tests carried out using models with different sizes (b¼60, 75, 90 mm) provided very similar results as far as subcritical and supercritical regimes are concerned. The transition phase was qualitatively similar, but appeared at a slightly different Re. 4. Discussion Two main issues clearly emerged during the experimentation described in Section 3 and deserve to be discussed. The former concerns the sharp discontinuity of the aerodynamic behavior observed for α ¼ αcr in smooth-flow condition. The latter issue is related to the two distinct flow behaviors observed for different Re in turbulent flow for r/b¼2/15. 4.1. Intermittent behavior at the critical angle of incidence For the case of sharp-corner square cylinders, the existence of two flow regimes separated by the critical angle of incidence has been clearly documented through accurate flow visualization techniques (Igarashi, 1984; Huang et al., 2010; Yen and Yang, 2011). The qualitative similarity of the behavior observed for the three tested models suggests that a modification of the corner geometry (within the considered limits) does not produce qualitative variations of the flow pattern, but rather modifies the limit of existence of the flow regimes known for the sharp corner case. In particular, the increment of r/b produces a reduction of αcr.
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2
0.8 0.6 0.4
1.5
CL
CD
0.2 0 -0.2
1
-0.4 -0.6 0.5 -10
-5
0
5
10
-0.8 -10
-5
α [°]
0
5
10
5
10
α [°]
0.3
0.7 0.6
0.25
St
0.5 0.2
0.4 0.3
0.15 0.2 0.1 -10
-5
0
5
10
0.1 -10
-5
α [°]
0
α [°]
Fig. 5. Steady drag (a) and lift (b) coefficients, Strouhal number (c) and fluctuating lift coefficient (d) for r/b ¼ 2/15 in turbulent flow as functions of α; effect of Re in the range 4.9 104–1.8 105.
Fig. 6. Steady pressure coefficients in smooth flow. Re¼2.9 104.
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Fig. 7. Wavelet map and time history of the lift force at α¼ αcr in smooth flow. Sharp-corner (a), rounded-corner with r/b ¼1/15 (b) and 2/15 (c). Re ¼4.7 104.
The transition between these two regimes is due to the reattachment of the mean flow on the lateral face exposed to the wind, and the consequent generation of a separation bubble. This behavior is confirmed by the analysis of the pressure field measured on the mid-span cross section of the model. Fig. 6 shows the mean pressure distribution acting on the surface of the three models in smooth flow for the angles of incidence 01, 51, 71, 121 and 151. Rotating the prism from α ¼01 to α ¼51 the mean pressure remains substantially unchanged and only a slight loss of symmetry is visible for the case r/b ¼2/15 for
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which α is approaching αcr. Rotating the prism further to α ¼71 the pressure field on the lateral face exposed to the wind of the model with r/b¼ 2/15 changes significantly. The suction increases near the leading edge and decreases in a small portion of the face close to the trailing edge, testifying the reattachment of the flow and the generation of a separation bubble. A similar modification is visible on the model with r/b¼1/15 for α passing from 71 to 121 and on the sharp-edge model for α increasing from 121 to 151. Once the flow reattaches on the lateral face exposed to the wind, further increments of the angle of incidence produce a shrinking of the separation bubble towards the windward corner. This effect is well recognizable in particular for the case r/b¼2/15. Fig. 7 shows a time–frequency analysis of the lift force measured in smooth flow at α ¼ αcr for the sharp-corner (a) and rounded-corner models with r/b¼1/15 (b) and 2/15 (c). The colormaps represent the amplitude of the wavelet transform of L, plotted with respect to the time t and the frequency n non-dimensionalized through b and U. The mother wavelet used for the analysis is an analytic Morlet type; the time histories of the lift coefficient are reported below the maps. The length of the considered time windows is 800 non-dimensional time units, which roughly correspond to 120 cycles of vortex shedding. From the wavelet maps it can be observed that the vortex-shedding peak frequency changes with time fluctuating between the values reported in Fig. 2 for angles just before and after αcr. This result suggests that the wavelet maps can be employed to study, from a qualitative point of view, the stability of the two concurrent flow regimes (subcritical and supercritical) that appear in the neighborhood of αcr, as well as the transition from one regime to the other. The comparison of the three wavelet maps reveals that the flow past the sharp-corner cylinder tends to have smooth and frequent transitions between the two regimes giving rise to a behavior that may be classified as irregular vortex shedding (Fig. 7(a)). On the contrary, in the case of rounded-corner cylinders a proper intermittent behavior appears as documented in Fig. 7(b) and (c), where the transition is clearly visible and the flow remains stable in a regime for a relatively long time (for example in Fig. 7(c) the length of the subcritical phase corresponds to about 50 cycles of vortex shedding). The comparison between Fig. 7(b) and (c) suggests that as r/b increases, the transition between the two regimes becomes sharper. A similar analysis has been carried out on the measurements obtained in turbulent flow, but no intermittence has been found. On the contrary, the transition through the critical angle appears quite smooth and characterized only by a reduction of the intensity of vortex shedding. 4.2. Inversion of the lift slope at small angle of incidence The analysis of the results obtained in turbulent flow for the rounded-corner cylinder with r/b¼2/15 suggests the existence of two flow regimes for small angle of incidence, whose transition is governed by Re. The most evident characteristics of the transition between the two regimes is the inversion of the slope of steady lift coefficient for α ¼01 that passes from negative to positive with obvious implications for the stability of cross-wind oscillations in the quasi-steady limit. This phenomenon is accompanied by the reduction of CD and the increment of St. A similar behavior has been experimentally observed on elongated rectangular cylinders with aspect ratio 3:1 and has been related to the reattachment of the flow on both the lateral faces of the body (Norberg, 1993). The occurrence of the steady flow reattachment is regulated by Re and by the characteristics of the free-stream turbulence (Nakamura and Ozono, 1987; Li and Melbourne, 1995). For the 3:1 rectangle the transition between separated flow and reattachment-like behavior has been identified between Re¼5 103 and 104 (Re defined with respect to the cross-flow dimension) in smooth flow condition (Norberg, 1993), but several evidences suggest that it can be anticipated by the addition of small-scale free-steam turbulence or by the transverse vibration of the body (Nakamura and Ohya, 1984; Nakamura and Hirata, 1989). It should be noted that the mentioned transition phenomenon is probably very sensitive to several flow characteristics including the scale of the freestream turbulence, the blockage ratio and possibly other parameters. Indeed, Larose and D'Auteuil (2008) observed a separated-like behavior on a 3:1 rectangle at Re ¼2.0 106 (Re defined with respect to the cross-flow dimension) in smooth flow, which turned into a reattachment-like behavior by adding 5% free-stream turbulence. Besides, they detected some unexplained effects of the Mach number. Fig. 8 shows the mean pressure coefficient measured on the rounded-corner model with r/b¼ 2/15 in turbulent flow at supercritical Re (between 105 and 2 105) for different angle of incidence. For α ¼ 0, as Re increases the separation bubbles shrink towards the leading edges anticipating the recovery of the base pressure, while the suction near the leading edges increases. The rotation of the cylinder tends to prevent the flow reattachment on the upper lateral face reaching, eventually,
α=0°
α=2°
α=4°
α=7°
α=10°
2 r/b=2/15
Fig. 8. Steady pressure coefficient in supercritical Re regime for r/b ¼2/15 in turbulent flow.
Re 1.01×105 1.15×105 1.28×105 1.42×105 1.54×105 1.68×105 1.81×105
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Fig. 9. Wavelet map and time history of the lift force for r/b ¼2/15, at Re¼ 1.42 105 in turbulent flow; α ¼ 21 (a), α ¼51 (b), α¼ 101 (c).
a flow configuration with fully-separated flow on the upper lateral face and reattachment-like flow on the lower lateral face (exposed to the wind). This flow condition is similar to the one observed for subcritical Re and α 4 αcr; this behavior has already been observed on elongated rectangular cylinders (Schewe, 2013) and is confirmed by the coincidence of CL and St for the two mentioned conditions (Fig. 5), even if some difference in the strength of vortex shedding exists, which is reflected into a difference of CD and C~ L .
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As Re increases, the maximum value of CL increases because the separation bubble on the upper lateral face can be sustained for larger angles of incidence. Once the separation bubble becomes unstable, CL decreases gradually to values typical of subcritical regime. During this transition, in which CL has a negative slope, no regular vortex shedding appears, probably due to the recurrent formation of an unstable separation bubble on the upper face. This behavior is documented in Fig. 9 where wavelet maps and the time histories of the lift coefficient are shown for different angles of incidence at Re¼1.42 105 in turbulent flow. For α ¼21 (a) the lift slope is positive and a regular vortex shedding appears with St about 0.26 (for α ¼01 the situation is similar); for α ¼51 (b) the lift slope is negative and no regular vortex shedding is present; for α ¼101 (c) regular vortex shedding reappears with St about 0.18. 5. Conclusions The effects of corner shaping on the aerodynamic behavior of square cylinders have been investigated through the analysis and discussion of wind-tunnel experiments. It has been found that rounded corners promote the reattachment of the flow on the lateral faces producing some significant effects that can be summarized as follows: 1. Rounded corners produces a reduction of the critical angle of incidence αcr for which the flow reattaches on the lateral face exposed to the wind. 2. The transition between subcritical and supercritical angle regime is sharp in smooth flow and gradual in turbulent flow; the sharp transition is accompanied by an intermittent behavior that in the sharp-corner case is not visible. 3. Rounded corners can produce the reattachment of the flow on the lateral faces for α ¼01, exactly as it happen for the case elongated rectangles. The effects of this flow-filed variation are confined in a range of relatively small angles of incidence and are characterized by the inversion of the lift slope and the strong increment of St. 4. The transition between subcritical and supercritical Re regime appeared very sensitive to the test conditions. Surely the scale of the free-stream turbulence plays a role, but probably other test parameters such as the blockage rate are important. These issues are currently under investigation. 5. The supercritical Re regime has been observed only for r/b¼2/15 in turbulent flow, however it cannot be excluded that it may appear also for smaller r/b ratios and smooth flow at sufficiently high Reynolds numbers. The technical implications of the mentioned effects are important due to the large variation of the aerodynamic coefficients as functions of r/b and possibly Re, as well as in relation to galloping instability. From the results obtained in smooth flow it can be concluded that the classical galloping model (based upon a quasi-steady assumption) cannot be adopted in the neighborhood of α ¼ αcr due to the observed intermittent behavior. On the other hand, the results obtained in turbulent flow reveal that the lift coefficient of rounded corner cylinders can be negative or positive for different Re and that, in supercritical Re regime, the necessary condition for galloping (negative lift slope) does not appear at α ¼0, but in a range of α between 51 and 101. References Delany, N.K., Sorensen, N.E., 1953. Low-Speed Drag of Cylinders of Various Shapes. Technical note 3038. National Advisory Committee for Aeronautics. Huang, R.F., Lin, B.H., Yen, S.C., 2010. Time-averaged topological flow patterns and their influence on vortex shedding of a square cylinder in crossflow at incidence. Journal of Fluids and Structures 26, 406–429. Huang, R.F., Lin, B.H., 2011. Effect of flow patterns on aerodynamic forces of a square cylinder at incidence. Journal of Mechanics 27, 347–355. Igarashi, T., 1984. Characteristics of the flow around a square prism. Bulletin of JSME 27, 1858–1865. Kwok, K.C.S., Wilhelm, P.A., Wilkie, B.G., 1988. Effect of edge configuration on wind-induced response of tall buildings. Engineering Structures 10, 135–140. Larose, G.L., D'Auteuil, A., 2008. Experiments on 2D rectangular prisms at high Reynolds numbers in a pressurised wind tunnel. Journal of Wind Engineering and Industrial Aerodynamics 96, 923–933. Lee, B.E., 1975. The effect of turbulence on the surface pressure field of a square prism. Journal of Fluid Mechanics 69, 321–352. Li, Q.S., Melbourne, W.H., 1995. An experimental investigation of the effects of free-stream turbulence on streamwise surface pressures in separated and reattaching flows. Journal of Wind Engineering and Industrial Aerodynamics 54–55, 313–323. Nakamura, Y., Ohya, Y., 1984. The effects of turbulence on the mean flow past two-dimensional rectangular cylinders. Journal of Fluid Mechanics 149, 255–273. Nakamura, Y., Ozono, S., 1987. The effects of turbulence on a separated and reattaching flow. Journal of Fluid Mechanics 178, 477–490. Nakamura, Y., Hirata, K., 1989. Critical geometry of oscillating bluff bodies. Journal of Fluid Mechanics 208, 375–393. Norberg, C., 1993. Flow around rectangular cylinders: pressure force and wake frequencies. Journal of Wind Engineering and Industrial Aerodynamics 49, 187–196. Okajima, A., 1982. Strouhal numbers of rectangular cylinders. Journal of Fluid Mechanics 123, 379–398. Schewe, G., 2013. Reynolds-number-effects in flow around a rectangular cylinder with aspect ratio 1:5. Journal of Fluids and Structures 39, 15–26. Tamura, T., Miyagi, T., Kitagishi, T., 1998. Numerical prediction of unsteady pressures on a square cylinder with various corner shapes. Journal of Wind Engineering and Industrial Aerodynamics 74–76, 531–542. Tamura, T., Miyagi, T., 1999. The effect of turbulence on aerodynamic forces on a square cylinder with various corner shapes. Journal of Wind Engineering and Industrial Aerodynamics 83, 135–145. Vickery, B.J., 1966. Fluctuating lift and drag on a long cylinder of square cross-section in a smooth and in a turbulent stream. Journal of Fluid Mechanics 25, 481–494. Yen, S.C., Yang, C.W., 2011. Flow patterns and vortex shedding behavior behind a square cylinder. Journal of Wind Engineering and Industrial Aerodynamics 99, 868–878.
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Journal of Fluids and Structures journal homepage: www.elsevier.com/locate/jfs
Experimental investigation on the aerodynamic behavior of square cylinders with rounded corners Luigi Carassale n, Andrea Freda, Michela Marrè-Brunenghi DICCA – Department of Civil, Chemical and Environmental Engineering, University of Genova, Italy
a r t i c l e i n f o
abstract
Article history: Received 8 January 2013 Accepted 10 October 2013 Available online 16 November 2013
The influence of corner shaping on the aerodynamic behavior of square cylinders is studied through wind tunnel tests. Beside the sharp-edge corner condition considered as a benchmark, two different rounded-corner radii (r/b¼ 1/15 and 2/15) are studied. Global forces and surface pressure are simultaneously measured in the Reynolds number range between 1.7 104 and 2.3 105. The measurements are extended to angles of incidence between 01 and 451, but the analysis and the discussion presented herein is focused on the low angle of incidence range. It is found that the critical angle of incidence, corresponding to the flow reattachment on the lateral face exposed to the flow, decreases as r/b increases and that an intermittent flow condition exists. In the case of turbulent incoming flow, two different aerodynamic regimes governed by the Reynolds number have been recognized. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Square cylinders Rounded corners Intermittence Critical angle Reynolds number effect
1. Introduction A careful modeling of the corners shape of building and structural elements has become a major objective for a windresponse-oriented optimal design. The introduction of rounded or chamfered corners has often the positive effect of reducing the drag force and the fluctuation of the transversal force due to vortex shedding (e.g. Kwok et al., 1988; Tamura and Miyagi, 1999); on the other hand, it can produce a more complicated aerodynamic behavior whose physical or numerical modeling may be challenging at design stage for wind engineers. As a matter of fact, in contrast to sharp-edge bodies, the rounding or the chamfering of the corners may lead to the absence of fixed separation points, and a significant dependency on both the Reynolds number and the characteristics of the incoming flow (e.g. Delany and Sorensen, 1953; Tamura and Miyagi, 1999; Tamura et al., 1998; Larose and D'Auteuil, 2008). The square cylinder is probably the simplest sharp-edge body and probably the most commonly investigated in aerodynamics; it is therefore the natural candidate to investigate the effect of corner shaping on the aerodynamic behavior of bluff bodies. With the twofold purpose of investigating the basic behavior of rounded corners and providing technical information useful for wind engineers, a wind-tunnel tests campaign has been carried out on square cylinders with rounded corners. Beside the sharp-corner case, rigid models of cylinders with two corner radii (r/b¼1/15 and 2/15) have been realized and tested. The global forces and the pressure field along the mid-span cross-section have been measured in the Reynolds number range between 1.7 104 and 2.3 105. Two levels of turbulence intensity (0.2% and 5%) have been considered. Section 2 provides a brief review of the current knowledge on the aerodynamic behavior of sharp-edge square cylinders, with particular reference to the qualitative modification of the flow field with the variation of angles of incidence.
n
Corresponding author. Tel.: þ39 010 353 2226. E-mail address: [email protected] (L. Carassale).
0889-9746/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfluidstructs.2013.10.010
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These concepts are then used as a guide for the interpretation of the experimental results presented in Section 3 and discussed in Section 4. The discussion is focused on two specific phenomena: (1) the intermittence of the vortex-shedding regime at the critical angle of incidence in smooth flow and (2) the transition from subcritical to supercritical Reynolds number regime in turbulent flow, with the inversion of the steady lift coefficient slope at zero angle of incidence.
2. Aerodynamic behavior of sharp-corner square cylinders: the dependency on the angle of incidence Sharp-edge square cylinders have been studied by several researchers who focused their attention on numerous aerodynamic parameters including pressure distribution, drag and lift forces, vortex shedding properties, as well as near wake velocity field (e.g. Vickery, 1966; Lee, 1975; Okajima, 1982; Igarashi, 1984; Nakamura and Ohya, 1984). The flow pattern around a square cylinder is strongly dependent on the angle of incidence α, and at least two characteristic flow regimes are clearly identified and separated by a critical angle αcr located about 12–151 (Igarashi, 1984). The boundary layer is completely separated from both the lateral faces for α o αcr, whereas the flow reattaches on the side exposed to the wind forming a separation bubble for α 4 αcr (e.g. Huang et al., 2010; Huang and Lin, 2011). A further sub-classification of the two regimes could be mentioned (Igarashi, 1984), but is not relevant in the present context. The subcritical regime (α o αcr) is characterized by negative slope of the lift coefficient, which sharply changes to positive as α becomes greater than αcr. Besides, the transition from the subcritical to the supercritical regime produces a rapid increment of the Strouhal number that corresponds to the reduction of the wake width due to the flow reattachment (Lee, 1975). In the critical regime both the drag coefficient and the fluctuating lift coefficient have a minimum value. This scenario is somehow dependent on the Reynolds number (Yen and Yang, 2011), as well as on the characteristics of the incoming flow. In particular, the thickening of the shear layers due to a small-scale free-stream turbulence promotes the formation of the separation bubble, which tends to appear for smaller angle of incidence and to shrink towards the leading edge (Lee, 1975).
3. Experimental results The experimental tests have been carried out in the closed-circuit wind tunnel at the University of Genova, whose cross section is 1640 1350 mm2. The models have span length l ¼500 mm; they are fabricated through the assemblage of aluminum plates and machined to reduce geometrical imperfections below 0.1% of the cross-section size b. The angle of incidence of the prisms is measured through a digital protractor with resolution 0.11. End plates are installed at the extremities of the models. A force balance realized by six resistive load cells is employed to measure global forces. The midspan cross section of the models is instrumented by a ring of N pressure taps (N ranges from 20 to 44 for different models) connected through short tubes to pressure scanners mounted inside the model. Beside the sharp-edge square cross section used as a benchmark test, two rounded-corner configurations with r/b¼1/15 and 2/15 (r being the corner radius) are considered. Fig. 1 shows the cross section of the tested cylinders and the reference system used for the presentation of the results. The force balance measurements are analyzed calculating the steady aerodynamic drag and lift coefficients (CD and CL), the Strouhal number (St) and the fluctuating lift coefficient (C~ L ) defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E½L2 E½L2 E½D E½L ns b ; C~ L ¼ CD ¼ ; CL ¼ ; St ¼ ; ð1Þ 2 2 U 0:5ρblU 0:5ρblU 0:5ρblU 2 where D and L are, respectively, the measured drag and lift forces (Fig. 1); ρ is the air density estimated on the basis of the temperature measured inside the wind tunnel; U is the free stream mean velocity measured through a static-Pitot tube installed upstream the test section; E[] is the statistic average operator that is implemented as a time average adopting the hypothesis of ergodic behavior; ns is the vortex shedding peak frequency. The vortex shedding frequency is estimated by fitting the power spectral density function of L through a Gaussian function in the neighborhood of the vortex shedding peak. The spectral analysis of L was carried out separately for the two concurrent flow configurations when a clear intermittent behavior was observed. Similar analyses carried out on the velocity recorded in the wake were used for verification purpose.
Fig. 1. Experimental setup.
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Analogously, the steady pressure coefficient is defined as Cp ¼
E½p p0 0:5ρU 2
;
ð2Þ
where p is the pressure measured on the body surface and p0 is the static wind-tunnel pressure. The influence of the Reynolds number is investigated varying both the wind-tunnel velocity U in the range 5–25 m/s, and the body size b in the range 50–150 mm. The Reynolds number is defined as Re ¼Ub/ν, ν being the kinematic viscosity, and considering the cross-section size as reference also for rotated configurations. The kinematic viscosity was calculated on the basis of the air temperature measured in the test section. No blockage correction is adopted since the primary purpose of the paper is the comparative analysis of the aerodynamic behavior of bodies with similar shape and the qualitative description of some observed fluid-dynamic phenomena. Besides, since these phenomena are related to strong modifications of the flow topology, the use of blockage-correction rules should be necessarily restricted to flow configurations that are far enough from transition points. This would introduce artificial discontinuities in the experimental data that would complicate interpretation. On the other hand, it is important to mention that several models with different size (corresponding to the ratios b/B from 2.5% to 11%, B being the size of the wind tunnel cross-section in direction orthogonal to the prism axis) have been tested and it emerged that the quality of the observed phenomena remains very consistent. Two conditions of incoming flow are considered: (1) smooth flow characterized by a longitudinal turbulence intensity Iu about 0.2% and (2) turbulent flow, produced through a grid realized by square bars, characterized by Iu about 5% and integral length scale Lu about 20 mm. Fig. 2 shows the variation of CD (a), CL (b), St (c) and C~ L (d) with respect to the angle of incidence for the three considered models in smooth flow. No significant influence of Re is found in the whole range explored during the experiments. It can be observed that these three models have a similar qualitative behavior characterized by the inversion of the lift slope at the critical angle of incidence αcr. For α ¼ αcr the steady drag and the fluctuating lift coefficients have a minimum value, while St increases sharply. The smooth fitting of the experimental data is interrupted at α ¼ αcr where a discontinuity exists 2.4
0.4
2.2
0.2 0 -0.2
CL
CD
2 1.8
-0.4 1.6
-0.6
1.4 1.2
-0.8 0
10
20
30
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50
-1
0
10
α [°]
20
30
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30
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α [°]
0.18
1.2
0.17
1
0.16
0.8
0.15
St
0.6 0.14 0.4
0.13
0.2
0.12 0.11
0
10
20
α [°]
30
40
50
0
0
10
20
α [°]
Fig. 2. Steady drag (a) and lift (b) coefficients, Strouhal number (c) and fluctuating lift coefficient (d) in smooth flow as functions of α. Re¼3.7 104 for r ¼0; Re¼ 2.7 104 for r/b¼ 1/15 and r/b¼ 2/15.
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2.4
0.8 0.6
2.2
0.4 0.2
1.8
CL
CD
2
0 -0.2
1.6
-0.4
1.4
-0.6 1.2 1
-0.8 0
10
20
30
40
50
-1
0
10
α [°]
20
30
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30
40
50
α [°] 1.2
0.2 0.19
1
0.18 0.8
St
0.17 0.16
0.6
0.15 0.4
0.14 0.13
0.2
0.12 0.11
0
10
20
α [°]
30
40
50
0
0
10
20
α [°]
Fig. 3. Steady drag (a) and lift (b) coefficients, Strouhal number (c) and fluctuating lift coefficient (d) in turbulent flow as functions of α. Re¼3.6 104 for r¼ 0; Re¼7.9 104 for r/b¼ 1/15; Re¼2.5 104 for r/b ¼ 2/15.
(see Section 4 for a discussion about this issue). The value of αcr decreases as r/b increases; for the sharp-corner model αcr has been found to be about 121, which is in substantial accord to previous experimentations (see Huang and Lin, 2011, for a review of previous results). In the case of rounded-corner models αcr is about 71 and 51 for r/b ¼1/15 and 2/15, respectively. Tamura and Miyagi (1999), working in similar flow conditions, found αcr ¼41 for r/b¼2.5/15. Fig. 3 shows the variation of CD (a), CL (b), St (c) and C~ L with respect to the angle of incidence, in turbulent flow at Re¼3.6 104 for r ¼0, Re¼7.9 104 for r/b¼ 1/15 and Re¼ 2.5 104 for r/b¼ 2/15. The qualitative comparison with the results presented in Fig. 2 shows that the increment of the free stream turbulence produces a slight reduction of the critical angle (for all the models) and promotes a smooth transition from subcritical to supercritical angle regimes. As far as the sharp-corner model and the rounded-corner model with r/b¼1/15 are concerned, the quantities shown in Fig. 3 are practically invariant through the whole observed range of Re (Re ¼1.7 104–1.6 105 for sharp-corner model and Re¼2.5 104–2.3 105 for r/b¼1/15). On the contrary, the model with r/b¼2/15 showed a strong dependency on Re, in particular for small angles of incidence. Fig. 4 shows CD (a), dCL/dα (b), St (c) and C~ L (d) measured in turbulent flow for α ¼ 0 on the rounded-corner model with r/b¼ 2/15. It can be observed that a drag crisis appears for Re between 5 104 and 1.2 105. Across this critical regime the drag coefficient drops from about 1.25 to 0.80; the lift slope passes from about 10 to þ 5 reaching values about 15 during the transition; St practically doubles jumping from 0.13 to 0.25; the fluctuating lift decreases significantly during the transition. Between Re¼6.6 104 and 8.6 104 no regular vortex shedding was observed. Delany and Sorensen (1953) carried out experiments on rounded-corner square cylinders in smooth flow finding a drag crisis at Re⋍6 105 for r/b¼2.5/15 and Re⋍2.5 105 for r/b¼5/15 (in the mentioned reference Re is defined consistently with the definition adopted herein). Fig. 5 shows the variation of CD (a), CL (b), St (c) and C~ L (d) with respect to the angle of incidence, for r/b¼2/15, in turbulent flow at different Re between 4.90 104 and 1.81 105. The transition observed in Fig. 4 reflects into the existence of two distinct behaviors referred to as subcritical (Re oRecr⋍7.5 104) and supercritical regime (Re4Recr). In the subcritical regime the drag coefficient for α ¼ 0 is about 1.3, (slightly lower than in smooth flow). The minimum value of the steady lift coefficient is about 0.7, and is obtained for α ¼3–41 (slightly before than αcr in smooth flow). The Strouhal number is about 0.13 for α ¼ 0 and rapidly increases up to 0.18 for α ¼41 (again not far from the values observed in smooth flow). As Re increases, the steady lift slope for α ¼0 switches from negative to positive. Further increments of Re have the result of extending the range of α where the slope is positive. The maximum value of the steady lift coefficient increases as
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10
1.4 1.3
5
dCL/dα
CD
1.2 1.1 1
0
-5
0.9 -10 0.8 0.7
0
0.5
1
1.5
2
x 105 2.5
-15
0
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1
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1.5
2
x 105 2.5
1.5
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x 105 2.5
Re 0.35
0.3
0.3
0.25
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0.15
St
0.35
0.1
0
0.5
1
1.5
2
x 105 2.5
Re
0.1
0
0.5
1
Re
Fig. 4. Steady drag (a) and lift (b) coefficients, Strouhal number (c) and fluctuating lift coefficient (d) for r/b¼ 2/15 in turbulent flow at α ¼ 01 as functions of Re.
Re increases and is obtained for larger and larger angles of incidence. The maximum observed CL is about 0.45 and appears for α ¼41 at Re¼1.81 105. In the region where the steady lift slope is positive St is about 0.27 (approximately twice the value obtained in smooth flow) and is practically insensitive to Re. As the slope of CL becomes negative, the regular vortex shedding disappears, until both CL and St recover the trend observed in subcritical regime; in the observed range of Re it happens before α ¼101. The minimum value of the drag coefficient is reached for the angle of incidence corresponding to the maximum values for CL. The results shown in Figs. 4 and 5 are obtained using a model with b¼150 mm. The tests carried out using models with different sizes (b¼60, 75, 90 mm) provided very similar results as far as subcritical and supercritical regimes are concerned. The transition phase was qualitatively similar, but appeared at a slightly different Re. 4. Discussion Two main issues clearly emerged during the experimentation described in Section 3 and deserve to be discussed. The former concerns the sharp discontinuity of the aerodynamic behavior observed for α ¼ αcr in smooth-flow condition. The latter issue is related to the two distinct flow behaviors observed for different Re in turbulent flow for r/b¼2/15. 4.1. Intermittent behavior at the critical angle of incidence For the case of sharp-corner square cylinders, the existence of two flow regimes separated by the critical angle of incidence has been clearly documented through accurate flow visualization techniques (Igarashi, 1984; Huang et al., 2010; Yen and Yang, 2011). The qualitative similarity of the behavior observed for the three tested models suggests that a modification of the corner geometry (within the considered limits) does not produce qualitative variations of the flow pattern, but rather modifies the limit of existence of the flow regimes known for the sharp corner case. In particular, the increment of r/b produces a reduction of αcr.
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2
0.8 0.6 0.4
1.5
CL
CD
0.2 0 -0.2
1
-0.4 -0.6 0.5 -10
-5
0
5
10
-0.8 -10
-5
α [°]
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5
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5
10
α [°]
0.3
0.7 0.6
0.25
St
0.5 0.2
0.4 0.3
0.15 0.2 0.1 -10
-5
0
5
10
0.1 -10
-5
α [°]
0
α [°]
Fig. 5. Steady drag (a) and lift (b) coefficients, Strouhal number (c) and fluctuating lift coefficient (d) for r/b ¼ 2/15 in turbulent flow as functions of α; effect of Re in the range 4.9 104–1.8 105.
Fig. 6. Steady pressure coefficients in smooth flow. Re¼2.9 104.
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Fig. 7. Wavelet map and time history of the lift force at α¼ αcr in smooth flow. Sharp-corner (a), rounded-corner with r/b ¼1/15 (b) and 2/15 (c). Re ¼4.7 104.
The transition between these two regimes is due to the reattachment of the mean flow on the lateral face exposed to the wind, and the consequent generation of a separation bubble. This behavior is confirmed by the analysis of the pressure field measured on the mid-span cross section of the model. Fig. 6 shows the mean pressure distribution acting on the surface of the three models in smooth flow for the angles of incidence 01, 51, 71, 121 and 151. Rotating the prism from α ¼01 to α ¼51 the mean pressure remains substantially unchanged and only a slight loss of symmetry is visible for the case r/b ¼2/15 for
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which α is approaching αcr. Rotating the prism further to α ¼71 the pressure field on the lateral face exposed to the wind of the model with r/b¼ 2/15 changes significantly. The suction increases near the leading edge and decreases in a small portion of the face close to the trailing edge, testifying the reattachment of the flow and the generation of a separation bubble. A similar modification is visible on the model with r/b¼1/15 for α passing from 71 to 121 and on the sharp-edge model for α increasing from 121 to 151. Once the flow reattaches on the lateral face exposed to the wind, further increments of the angle of incidence produce a shrinking of the separation bubble towards the windward corner. This effect is well recognizable in particular for the case r/b¼2/15. Fig. 7 shows a time–frequency analysis of the lift force measured in smooth flow at α ¼ αcr for the sharp-corner (a) and rounded-corner models with r/b¼1/15 (b) and 2/15 (c). The colormaps represent the amplitude of the wavelet transform of L, plotted with respect to the time t and the frequency n non-dimensionalized through b and U. The mother wavelet used for the analysis is an analytic Morlet type; the time histories of the lift coefficient are reported below the maps. The length of the considered time windows is 800 non-dimensional time units, which roughly correspond to 120 cycles of vortex shedding. From the wavelet maps it can be observed that the vortex-shedding peak frequency changes with time fluctuating between the values reported in Fig. 2 for angles just before and after αcr. This result suggests that the wavelet maps can be employed to study, from a qualitative point of view, the stability of the two concurrent flow regimes (subcritical and supercritical) that appear in the neighborhood of αcr, as well as the transition from one regime to the other. The comparison of the three wavelet maps reveals that the flow past the sharp-corner cylinder tends to have smooth and frequent transitions between the two regimes giving rise to a behavior that may be classified as irregular vortex shedding (Fig. 7(a)). On the contrary, in the case of rounded-corner cylinders a proper intermittent behavior appears as documented in Fig. 7(b) and (c), where the transition is clearly visible and the flow remains stable in a regime for a relatively long time (for example in Fig. 7(c) the length of the subcritical phase corresponds to about 50 cycles of vortex shedding). The comparison between Fig. 7(b) and (c) suggests that as r/b increases, the transition between the two regimes becomes sharper. A similar analysis has been carried out on the measurements obtained in turbulent flow, but no intermittence has been found. On the contrary, the transition through the critical angle appears quite smooth and characterized only by a reduction of the intensity of vortex shedding. 4.2. Inversion of the lift slope at small angle of incidence The analysis of the results obtained in turbulent flow for the rounded-corner cylinder with r/b¼2/15 suggests the existence of two flow regimes for small angle of incidence, whose transition is governed by Re. The most evident characteristics of the transition between the two regimes is the inversion of the slope of steady lift coefficient for α ¼01 that passes from negative to positive with obvious implications for the stability of cross-wind oscillations in the quasi-steady limit. This phenomenon is accompanied by the reduction of CD and the increment of St. A similar behavior has been experimentally observed on elongated rectangular cylinders with aspect ratio 3:1 and has been related to the reattachment of the flow on both the lateral faces of the body (Norberg, 1993). The occurrence of the steady flow reattachment is regulated by Re and by the characteristics of the free-stream turbulence (Nakamura and Ozono, 1987; Li and Melbourne, 1995). For the 3:1 rectangle the transition between separated flow and reattachment-like behavior has been identified between Re¼5 103 and 104 (Re defined with respect to the cross-flow dimension) in smooth flow condition (Norberg, 1993), but several evidences suggest that it can be anticipated by the addition of small-scale free-steam turbulence or by the transverse vibration of the body (Nakamura and Ohya, 1984; Nakamura and Hirata, 1989). It should be noted that the mentioned transition phenomenon is probably very sensitive to several flow characteristics including the scale of the freestream turbulence, the blockage ratio and possibly other parameters. Indeed, Larose and D'Auteuil (2008) observed a separated-like behavior on a 3:1 rectangle at Re ¼2.0 106 (Re defined with respect to the cross-flow dimension) in smooth flow, which turned into a reattachment-like behavior by adding 5% free-stream turbulence. Besides, they detected some unexplained effects of the Mach number. Fig. 8 shows the mean pressure coefficient measured on the rounded-corner model with r/b¼ 2/15 in turbulent flow at supercritical Re (between 105 and 2 105) for different angle of incidence. For α ¼ 0, as Re increases the separation bubbles shrink towards the leading edges anticipating the recovery of the base pressure, while the suction near the leading edges increases. The rotation of the cylinder tends to prevent the flow reattachment on the upper lateral face reaching, eventually,
α=0°
α=2°
α=4°
α=7°
α=10°
2 r/b=2/15
Fig. 8. Steady pressure coefficient in supercritical Re regime for r/b ¼2/15 in turbulent flow.
Re 1.01×105 1.15×105 1.28×105 1.42×105 1.54×105 1.68×105 1.81×105
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Fig. 9. Wavelet map and time history of the lift force for r/b ¼2/15, at Re¼ 1.42 105 in turbulent flow; α ¼ 21 (a), α ¼51 (b), α¼ 101 (c).
a flow configuration with fully-separated flow on the upper lateral face and reattachment-like flow on the lower lateral face (exposed to the wind). This flow condition is similar to the one observed for subcritical Re and α 4 αcr; this behavior has already been observed on elongated rectangular cylinders (Schewe, 2013) and is confirmed by the coincidence of CL and St for the two mentioned conditions (Fig. 5), even if some difference in the strength of vortex shedding exists, which is reflected into a difference of CD and C~ L .
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As Re increases, the maximum value of CL increases because the separation bubble on the upper lateral face can be sustained for larger angles of incidence. Once the separation bubble becomes unstable, CL decreases gradually to values typical of subcritical regime. During this transition, in which CL has a negative slope, no regular vortex shedding appears, probably due to the recurrent formation of an unstable separation bubble on the upper face. This behavior is documented in Fig. 9 where wavelet maps and the time histories of the lift coefficient are shown for different angles of incidence at Re¼1.42 105 in turbulent flow. For α ¼21 (a) the lift slope is positive and a regular vortex shedding appears with St about 0.26 (for α ¼01 the situation is similar); for α ¼51 (b) the lift slope is negative and no regular vortex shedding is present; for α ¼101 (c) regular vortex shedding reappears with St about 0.18. 5. Conclusions The effects of corner shaping on the aerodynamic behavior of square cylinders have been investigated through the analysis and discussion of wind-tunnel experiments. It has been found that rounded corners promote the reattachment of the flow on the lateral faces producing some significant effects that can be summarized as follows: 1. Rounded corners produces a reduction of the critical angle of incidence αcr for which the flow reattaches on the lateral face exposed to the wind. 2. The transition between subcritical and supercritical angle regime is sharp in smooth flow and gradual in turbulent flow; the sharp transition is accompanied by an intermittent behavior that in the sharp-corner case is not visible. 3. Rounded corners can produce the reattachment of the flow on the lateral faces for α ¼01, exactly as it happen for the case elongated rectangles. The effects of this flow-filed variation are confined in a range of relatively small angles of incidence and are characterized by the inversion of the lift slope and the strong increment of St. 4. The transition between subcritical and supercritical Re regime appeared very sensitive to the test conditions. Surely the scale of the free-stream turbulence plays a role, but probably other test parameters such as the blockage rate are important. These issues are currently under investigation. 5. The supercritical Re regime has been observed only for r/b¼2/15 in turbulent flow, however it cannot be excluded that it may appear also for smaller r/b ratios and smooth flow at sufficiently high Reynolds numbers. The technical implications of the mentioned effects are important due to the large variation of the aerodynamic coefficients as functions of r/b and possibly Re, as well as in relation to galloping instability. From the results obtained in smooth flow it can be concluded that the classical galloping model (based upon a quasi-steady assumption) cannot be adopted in the neighborhood of α ¼ αcr due to the observed intermittent behavior. On the other hand, the results obtained in turbulent flow reveal that the lift coefficient of rounded corner cylinders can be negative or positive for different Re and that, in supercritical Re regime, the necessary condition for galloping (negative lift slope) does not appear at α ¼0, but in a range of α between 51 and 101. References Delany, N.K., Sorensen, N.E., 1953. Low-Speed Drag of Cylinders of Various Shapes. Technical note 3038. National Advisory Committee for Aeronautics. Huang, R.F., Lin, B.H., Yen, S.C., 2010. Time-averaged topological flow patterns and their influence on vortex shedding of a square cylinder in crossflow at incidence. Journal of Fluids and Structures 26, 406–429. Huang, R.F., Lin, B.H., 2011. Effect of flow patterns on aerodynamic forces of a square cylinder at incidence. Journal of Mechanics 27, 347–355. Igarashi, T., 1984. Characteristics of the flow around a square prism. Bulletin of JSME 27, 1858–1865. Kwok, K.C.S., Wilhelm, P.A., Wilkie, B.G., 1988. Effect of edge configuration on wind-induced response of tall buildings. Engineering Structures 10, 135–140. Larose, G.L., D'Auteuil, A., 2008. Experiments on 2D rectangular prisms at high Reynolds numbers in a pressurised wind tunnel. Journal of Wind Engineering and Industrial Aerodynamics 96, 923–933. Lee, B.E., 1975. The effect of turbulence on the surface pressure field of a square prism. Journal of Fluid Mechanics 69, 321–352. Li, Q.S., Melbourne, W.H., 1995. An experimental investigation of the effects of free-stream turbulence on streamwise surface pressures in separated and reattaching flows. Journal of Wind Engineering and Industrial Aerodynamics 54–55, 313–323. Nakamura, Y., Ohya, Y., 1984. The effects of turbulence on the mean flow past two-dimensional rectangular cylinders. Journal of Fluid Mechanics 149, 255–273. Nakamura, Y., Ozono, S., 1987. The effects of turbulence on a separated and reattaching flow. Journal of Fluid Mechanics 178, 477–490. Nakamura, Y., Hirata, K., 1989. Critical geometry of oscillating bluff bodies. Journal of Fluid Mechanics 208, 375–393. Norberg, C., 1993. Flow around rectangular cylinders: pressure force and wake frequencies. Journal of Wind Engineering and Industrial Aerodynamics 49, 187–196. Okajima, A., 1982. Strouhal numbers of rectangular cylinders. Journal of Fluid Mechanics 123, 379–398. Schewe, G., 2013. Reynolds-number-effects in flow around a rectangular cylinder with aspect ratio 1:5. Journal of Fluids and Structures 39, 15–26. Tamura, T., Miyagi, T., Kitagishi, T., 1998. Numerical prediction of unsteady pressures on a square cylinder with various corner shapes. Journal of Wind Engineering and Industrial Aerodynamics 74–76, 531–542. Tamura, T., Miyagi, T., 1999. The effect of turbulence on aerodynamic forces on a square cylinder with various corner shapes. Journal of Wind Engineering and Industrial Aerodynamics 83, 135–145. Vickery, B.J., 1966. Fluctuating lift and drag on a long cylinder of square cross-section in a smooth and in a turbulent stream. Journal of Fluid Mechanics 25, 481–494. Yen, S.C., Yang, C.W., 2011. Flow patterns and vortex shedding behavior behind a square cylinder. Journal of Wind Engineering and Industrial Aerodynamics 99, 868–878.